Bevel inside cylinder
Well, you are not only bevelling the two circular edge rings, you are bevelling every edge between them.

That can for example happen when one uses Vertex Select Mode. When selecting vertices, every edge between them is automatically selected, too. That's why it's not possible to extrude a checker pattern in one go. 



Use Edge Select Mode and only select the two edge rings, for example with ALT right click. Then you will get a nice and clean bevel. Make sure your object isn't scaled. That can distort the bevel as well.